On a decomposition of regular domains into John domains with uniform constants

Friedrich, Manuel

ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 4, pp. 1541-1583.

We derive a decomposition result for regular, two-dimensional domains into John domains with uniform constants. We prove that for every simply connected domain $\Omega \subset {ℝ}^2$ with $C^1$-boundary there is a corresponding partition $\Omega ={\Omega }_1\cup ...\cup {\Omega }_N$ with ${\Sigma }_{j=1}^N{\mathscr{H}}^1(\partial {\Omega }_j\setminus \partial \Omega )\le \theta$ such that each component is a John domain with a John constant only depending on $\theta$. The result implies that many inequalities in Sobolev spaces such as Poincaré’s or Korn’s inequality hold on the partition of $\Omega$ for uniform constants, which are independent of $\Omega$.

Classification : 26D10, 70G75, 46E35
Mots clés : John domains, Korn’s inequality, free discontinuity problems, shape optimization problems

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